4.3 Collisionless matter

A cloud of collisionless particles can be described by the Vlasov equation, i.e., the Boltzmann
equation without collision term. This matter model differs from field theories by having a much
larger number of matter degrees of freedom: The matter content is described by a statistical
distribution on the point particle phase space, instead of a finite number of fields
. When restricted to spherical symmetry, individual particles move tangentially as well as
radially, and so individually have angular momentum, but the stress-energy tensor averages
out to a spherically symmetric one, with zero total angular momentum. The distribution
is then a function of radius, time, radial momentum and (conserved) angular
momentum.

Several numerical simulations of critical collapse of collisionless matter in spherical symmetry have been
published to date, and remarkably no type II scaling phenomena has been discovered. Indications of type I
scaling have been found, but these do not quite fit the standard picture of critical collapse. Rein
et al. [183] find that black hole formation turns on with a mass gap that is a large part of
the ADM mass of the initial data, and this gap depends on the initial matter condition. No
critical behavior of either type I or type II was observed. Olabarrieta and Choptuik [168] find
evidence of a metastable static solution at the black hole threshold, with type I scaling of its
life time as in Equation (13). However, the critical exponent depends weakly on the family of
initial data, ranging from 5.0 to 5.9, with a quoted uncertainty of 0.2. Furthermore, the matter
distribution does not appear to be universal, while the metric seems to be universal up to an overall
rescaling, so that there appears to be no universal critical solution. More precise computations by
Stevenson and Choptuik [192], using finite volume HRSC methods, have confirmed the existence of
static intermediate solutions and non-universal scaling with exponents ranging now from 5.27 to
11.65.

Martín-García and Gundlach [155] have constructed a family of CSS spherically symmetric solutions
for massless particles that is generic by function counting. There are infinitely many solutions with different
matter configurations but the same stress-energy tensor and spacetime metric, due to the existence of an
exact symmetry: Two massless particles with energy-momentum in the solution can be replaced by one
particle with . A similar result holds for the perturbations. As the growth exponent of a
perturbation mode can be determined from the metric alone, this means that there are infinitely many
perturbation modes with the same . If there is one growing perturbative mode, there are infinitely many.
Therefore a candidate critical solution (either static or CSS) cannot be isolated or have only one growing
mode. This argument rules out the existence of both type I and type II critical phenomena (in their
standard form, i.e., including universality) for massless particles in the complete system, but
some partial form of criticality could still be found by restricting to sections of phase space in
which that symmetry is broken, for example by prescribing a fixed form for the dependence
of the distribution function f on angular momentum L, as those numerical simulations have
done.

A recent investigation of Andréasson and Rein [8] with massive particles has confirmed again the
existence of a mass gap and the existence of metastable static solutions at the black hole threshold, though
there is no estimation of the scaling of their life-times. More interestingly, they show that the subcritical
regime can lead to either dispersion or an oscillating steady state depending on the binding energy of the
system. They also conclude, based on perturbative arguments, that there cannot be an isolated universal
critical solution.

More numerical work is still required, but current evidence suggests that there are no type II critical
phenomena, and that there is a continuum of critical solutions in type I critical phenomena and hence only
limited universality.